Finite Group DatabasebetaFinite Group DatabasebetaOrder 3528.
152 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C22×C72 | C2×C32 |
| LCS | G ▷ C22×C72 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. ΓL(1,𝔽8)×C22⋊ΓL(1,𝔽8) | C2 | semidirect |
| 2. (C22⋊ΓL(1,𝔽8))⋊D7 | C3 | semidirect |
| 3. #11924339 | C3 | semidirect |
| 4. #11924338 | C3 | semidirect |
| 5. C22×ΓL(1,𝔽8)⋊D7 | C3 | semidirect |
| 7. C22⋊#3862 | C2×C3 | semidirect |
| 8. C22⋊#3861 | C2×C3 | semidirect |
| 9. C7×C22⋊ΓL(1,𝔽8) | C2×C3 | semidirect |
| 10. C22×C7×ΓL(1,𝔽8) | C2×C3 | semidirect |
| 16. C22×C7⋊D7 | C32 | semidirect |
| 22. ΓL(1,𝔽8)⋊D7 | A4 | semidirect |
| 25. C22×C72 | C2×C32 | semidirect |
| 40. C7×ΓL(1,𝔽8) | C2×A4 | semidirect |
| 45. C7⋊D7 | C3×A4 | semidirect |
| 54. C22⋊ΓL(1,𝔽8) | AGL(1,𝔽7) | semidirect |
| 56. C22×ΓL(1,𝔽8) | AGL(1,𝔽7) | semidirect |
| 65. C72 | C2×C3×A4 | semidirect |
| 81. C22×C7 | C3×AGL(1,𝔽7) | semidirect |
| 82. C22×C7 | C3×AGL(1,𝔽7) | semidirect |
| 101. ΓL(1,𝔽8) | D7⋊A4 | semidirect |
| 121. C7 | A4×AGL(1,𝔽7) | semidirect |
| 122. C7 | C3×D7⋊A4 | semidirect |
| 131. C22 | ΓL(1,𝔽8)⋊AGL(1,𝔽7) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×C22⋊ΓL(1,𝔽8) | 2 |
| (C22⋊ΓL(1,𝔽8))⋊D7 | 3 |
| #11924339 | 3 |
| #11924338 | 3 |
| C22×ΓL(1,𝔽8)⋊D7 | 3 |
| ΓL(1,𝔽8)⋊AGL(1,𝔽7) | 4 |
| C3×D7⋊A4 | 7 |
| A4×AGL(1,𝔽7) | 7 |